Editing Caesium standard (section)

== Parameters and significance in the second and other SI units ==
Suppose the caesium standard has the parameters:
* [[Speed of light|Velocity]]: ''c''
* [[Planck constant|Energy/frequency]]: ''h''
* Time period: {{math|Δ''t''<sub>Cs</sub>}}
* Frequency: {{math|Δ''ν''<sub>Cs</sub>}}
* Wavelength: {{math|Δ''λ''<sub>Cs</sub>}}
* Photon energy: {{math|Δ''E''<sub>Cs</sub>}}
* [[Mass–energy equivalence|Photon mass equivalent]]: {{math|Δ''M''<sub>Cs</sub>}} 

=== Time and frequency ===

The first set of units defined using the caesium standard were those relating to time, with the second being defined in 1967 as "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom" meaning that:
* 1 [[second]], s, = 9,192,631,770 {{math|Δ''t''<sub>Cs</sub>}}
* 1 [[hertz]], Hz, = 1/s = {{sfrac|{{math|Δ''ν''<sub>Cs</sub>}}|9,192,631,770}}
* 1 [[becquerel]], Bq, = 1 nuclear decay/s = {{sfrac|1|9,192,631,770}} nuclear decays/{{math|Δ''t''<sub>Cs</sub>}}

This also linked the definitions of the derived units relating to force and energy (see below) and of the ampere, whose definition at the time made reference to the newton, to the caesium standard. Before 1967 the SI units of time and frequency were defined using the [[tropical year]] and before 1960 by the length of the [[solar time|mean solar day]]<ref>{{Cite web|url=https://www.bipm.org/en/history-si/second|title = Second – BIPM}}</ref>

=== Length ===

In 1983, the meter was, indirectly, defined in terms of the caesium standard with the formal definition "The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. This implied:
* 1 [[metre]], m, = {{sfrac|''c s''|299,792,458}} = {{sfrac|9,192,631,770|299,792,458}} ''c'' {{math|Δ''t''<sub>Cs</sub>}} = {{sfrac|9,192,631,770|299,792,458}} {{math|Δ''λ''<sub>Cs</sub>}}
* 1 [[radian]], rad, = 1 m/m = {{math|Δ''λ''<sub>Cs</sub>}}/{{math|Δ''λ''<sub>Cs</sub>}} = 1 (dimensionless unit of angle)
* 1 [[steradian]], sr, = 1 m<sup>2</sup>/m<sup>2</sup> = {{math|Δ''λ''<sub>Cs</sub>}}<sup>2</sup>/{{math|Δ''λ''<sub>Cs</sub>}}<sup>2</sup> = 1 (dimensionless unit of [[solid angle]])

Between 1960 and 1983, the metre had been defined by the wavelength of a different transition frequency associated with the [[isotopes of krypton|krypton-86 atom]]. This had a much higher frequency and shorter wavelength than the caesium standard, falling inside the [[visible spectrum]]. The first definition, used between 1889 and 1960, was by the [[international prototype meter]].<ref>{{Cite web|url=https://www.bipm.org/en/history-si/metre|title = Metre - BIPM}}</ref>

=== Mass, energy, and force ===

Following the [[2019 revision of the SI]], electromagnetic radiation, in general, was explicitly defined to have the exact parameters:
* ''c'' = 299,792,458 m/s
* ''h'' = {{val|6.62607015|e=-34}} J s

The caesium-133 hyperfine transition radiation was explicitly defined to have frequency:
* {{math|Δ''ν''<sub>Cs</sub>}} = 9,192,631,770 Hz<ref>{{Cite web|url=https://www.bipm.org/en/committees/cg/cgpm/26-2018/resolution-1|title = Resolution 1 (2018) - BIPM}}</ref>

Though the above values for ''c'' and {{math|Δ''ν''<sub>Cs</sub>}} were already obviously implicit in the definitions of the metre and second. Together they imply:
* {{math|Δ''t''<sub>Cs</sub>}} = {{sfrac|1|{{math|Δ''ν''<sub>Cs</sub>}}}} = {{sfrac|s|9,192,631,770}}
* {{math|Δ''λ''<sub>Cs</sub>}} = ''c'' {{math|Δ''t''<sub>Cs</sub>}} = {{sfrac|299,792,458|9,192,631,770}} m
* {{math|Δ''E''<sub>Cs</sub>}} = ''h'' {{math|Δ''ν''<sub>Cs</sub>}} = 9,192,631,770 Hz × {{val|6.62607015|e=-34}} J s = {{val|6.09110229711386655|e=-24}} J
* {{math|Δ''M''<sub>Cs</sub>}} = {{sfrac|{{math|Δ''E''<sub>Cs</sub>}}|''c''<sup>2</sup>}} = {{sfrac|{{val|6.09110229711386655|e=-24}} J|89,875,517,873,681,764 m<sup>2</sup>/s<sup>2</sup>}} = {{sfrac|{{val|6.09110229711386655}}|{{val|8.9875517873681764|e=40}}}} kg

Notably, the wavelength has a fairly human-sized value of about 3.26 centimetres and the photon energy is surprisingly close to the average molecular [[kinetic energy]] per [[Degrees of freedom (physics and chemistry)|degree of freedom]] per [[kelvin]]. From these it follows that:
* 1 [[kilogram]], kg, = {{sfrac|{{val|8.9875517873681764|e=40}}|{{val|6.09110229711386655}}}} {{math|Δ''M''<sub>Cs</sub>}}
* 1 [[joule]], J, = {{sfrac|{{val|e=24}}|{{val|6.09110229711386655}}}} {{math|Δ''E''<sub>Cs</sub>}}
* 1 [[watt]], W, = 1 J/s = {{sfrac|{{val|e=14}}|{{val|5.59932604907689089550702935}}}} {{math|Δ''E''<sub>Cs</sub>}} {{math|Δ''ν''<sub>Cs</sub>}}
* 1 [[Newton (unit)|newton]], N, = 1 J/m = {{sfrac|{{val|2.99792458|e=22}}|{{val|5.59932604907689089550702935}}}} {{math|Δ''E''<sub>Cs</sub>}}/{{math|Δ''λ''<sub>Cs</sub>}}
* 1 [[Pascal (unit)|pascal]], Pa, = 1 N/m<sup>2</sup> = {{sfrac|{{val|2.6944002417373989539335912|e=19}}|{{val|4.73168129737820913189287698892486811451620615}}}} {{math|Δ''E''<sub>Cs</sub>}}/{{math|Δ''λ''<sub>Cs</sub>}}<sup>3</sup>
* 1 [[Gray (unit)|gray]], Gy, = 1 J/kg = {{sfrac|1|89,875,517,873,681,764}} {{math|Δ''E''<sub>Cs</sub>}}/{{math|Δ''M''<sub>Cs</sub>}} = {{sfrac|''c''<sup>2</sup>|89,875,517,873,681,764}}
* 1 [[sievert]], Sv, = the [[ionizing radiation]] dose [[Equivalent dose|equivalent]] to 1 gray of [[gamma rays]]

Prior to the revision, between 1889 and 2019, the family of metric (and later SI) units relating to mass, force, and energy were somewhat notoriously defined by the mass of the [[International Prototype of the Kilogram]] (IPK), a specific object stored at the headquarters of the [[International Bureau of Weights and Measures]] in [[Paris]], meaning that any change to the mass of that object would have resulted in a change to the size of the kilogram and of the many other units whose value at the time depended on that of the kilogram.<ref>{{Cite web|url=https://www.bipm.org/en/history-si/kilogram|title = Kilogram - BIPM}}</ref>

=== Temperature ===

From 1954 to 2019, the SI temperature scales were defined using the [[triple point]] of water and [[absolute zero]].<ref>{{Cite web|url=https://www.bipm.org/en/history-si/kelvin|title = Kelvin - BIPM}}</ref> The 2019 revision replaced these with an assigned value for the [[Boltzmann constant]], ''k'', of {{val|1.380649|e=-23}} J/K, implying:
* 1 [[kelvin]], K, = {{val|1.380649|e=-23}} J/2 per degree of freedom = {{sfrac|{{val|1.380649|e=-23}} × {{val|e=24}}/2|{{val|6.09110229711386655}}}} {{math|Δ''E''<sub>Cs</sub>}} per degree of freedom = {{sfrac|{{val|1.380649}}|{{val|1.21822045942277331}}}} {{math|Δ''E''<sub>Cs</sub>}} per degree of freedom
* Temperature in degrees [[Celsius]], °C, = temperature in kelvins − 273.15 = {{sfrac|{{val|1.21822045942277331}} × kinetic energy per degree of freedom − {{val|377.12427435}} {{math|Δ''E''<sub>Cs</sub>}}|{{val|1.380649}} {{math|Δ''E''<sub>Cs</sub>}}}}

=== Amount of substance ===

The mole is [[Avogadro constant|an extremely large number]] of "elementary entities" (i.e. [[atoms]], [[molecules]], [[ions]], etc). From 1969 to 2019, this number was 0.012 × the mass ratio between the IPK and a [[Isotopes of carbon|carbon 12 atom]].<ref>{{Cite web|url=https://www.bipm.org/en/history-si/mole|title=Mole - BIPM}}</ref> The 2019 revision simplified this by assigning the Avogadro constant the exact value {{val|6.02214076|e=23}} elementary entities per mole, thus, uniquely among the base units, the mole maintained its independence from the caesium standard:
* 1 [[Mole (unit)|mole]], mol, = {{val|6.02214076|e=23}} elementary entities
* 1 [[katal]], kat, = 1 mol/s = {{sfrac|{{val|6.02214076|e=14}}|{{val|9.19263177}}}} elementary entities/{{math|Δ''t''<sub>Cs</sub>}}

=== Electromagnetic units ===

Prior to the revision, the ampere was defined as the current needed to [[Ampère's force law|produce a force]] between 2 parallel wires 1 m apart of 0.2 [[Newton (unit)|μN]] per meter. The 2019 revision replaced this definition by giving [[elementary charge|the charge on the electron]], ''e'', the exact value {{val|1.602176634|e=-19}} coulombs. Somewhat incongruously, the coulomb is still considered a derived unit and the ampere a base unit, rather than vice versa.<ref>{{Cite web|url=https://www.bipm.org/en/history-si/ampere|title = Ampere - BIPM}}</ref> In any case, this convention entailed the following exact relationships between the SI electromagnetic units, elementary charge, and the caesium-133 hyperfine transition radiation:
* 1 [[coulomb]], C, = {{sfrac|{{val|e=19}}|{{val|1.602176634}}}} ''e''
* 1 [[ampere]], or amp, A, = 1 C/s = {{sfrac|{{val|e=9}}|{{val|1.472821982686006218}}}} ''e'' {{math|Δ''ν''<sub>Cs</sub>}}
* 1 [[volt]], V, = 1 J/C = {{sfrac|{{val|1.602176634|e=5}}|{{val|6.09110229711386655}}}} {{math|Δ''E''<sub>Cs</sub>}}/''e''
* 1 [[farad]], F, = 1 C/V = {{sfrac|{{val|6.09110229711386655|e=14}}|{{val|2.566969966535569956}}}} ''e''<sup>2</sup>/{{math|Δ''E''<sub>Cs</sub>}}
* 1 [[Ohm (unit)|ohm]], Ω, = 1 V/A = {{sfrac|{{val|2.359720966701071721258310212|e=-4}}|{{val|6.09110229711386655}}}} {{math|Δ''E''<sub>Cs</sub>}}/{{math|Δ''ν''<sub>Cs</sub>}} ''e''<sup>2</sup> = {{sfrac|{{val|2.359720966701071721258310212|e=-4}}|{{val|6.09110229711386655}}}} ''h''/''e''<sup>2</sup>
* 1 [[Siemens (unit)|siemens]], S, = 1/Ω = {{sfrac|{{val|6.09110229711386655|e=4}}|{{val|2.359720966701071721258310212}}}} ''e''<sup>2</sup>/''h''
* 1 [[Weber (unit)|weber]], Wb, = 1 V s = {{sfrac|{{val|1.602176634|e=15}}|{{val|6.62607015}}}} {{math|Δ''E''<sub>Cs</sub>}} {{math|Δ''t''<sub>Cs</sub>}}/''e'' = {{sfrac|{{val|1.602176634|e=15}}|{{val|6.62607015}}}} ''h''/''e''
* 1 [[Tesla (unit)|tesla]], T, = 1 Wb/m<sup>2</sup> = {{sfrac|{{val|1.43996454705862285832702376|e=12}}|{{val|5.59932604907689089550702935}}}} {{math|Δ''E''<sub>Cs</sub>}} {{math|Δ''t''<sub>Cs</sub>}}/''e'' {{math|Δ''λ''<sub>Cs</sub>}}<sup>2</sup> = {{sfrac|{{val|1.43996454705862285832702376|e=12}}|{{val|5.59932604907689089550702935}}}} ''E''/''e c'' {{math|Δ''λ''<sub>Cs</sub>}}
* 1 [[Henry (unit)|henry]], H, = Ω s = {{sfrac|{{val|2.359720966701071721258310212|e=6}}|{{val|6.62607015}}}} ''h'' {{math|Δ''t''<sub>Cs</sub>}}/''e''<sup>2</sup>

=== Optical units ===
From 1967 to 1979 the SI optical units, lumen, lux, and candela are defined using the [[Incandescence|incandescent]] glow of [[platinum]] at its melting point. After 1979, the candela was defined as the [[luminous intensity]] of a [[Monochromatic radiation|monochromatic]] [[Light|visible light]] source of frequency 540&nbsp;THz (i.e {{sfrac|6000|1.02140353}} that of the caesium standard) and [[radiant intensity]] {{sfrac|1|683}} watts per steradian. This linked the definition of the candela to the caesium standard and, until 2019, to the IPK. Unlike the units relating to [[mass]], [[energy]], [[temperature]], [[amount of substance]], and [[electromagnetism]], the optical units were ''not'' massively redefined in 2019, though they were indirectly affected since their values depend on that of the watt, and hence of the kilogram.<ref>{{Cite web|url=https://www.bipm.org/en/history-si/candela|title=Candela - BIPM}}</ref> The frequency used to define the optical units has the parameters:
* Frequency: 540&nbsp;THz
* Time period: {{sfrac|50|27}} [[femtosecond|fs]]
* Wavelength: {{sfrac|14.9896229|27}} [[micrometre|μm]]
* Photon energy: {{val|5.4|e=14}} Hz × {{val|6.62607015|e=-34}} J s = {{val|3.578077881|e=-19}} J
* [[luminous efficacy]], ''K''<sub>CD</sub>, = 683 lm/W
* [[luminous energy]] per photon, <math>Q_\mathrm v</math>, = {{val|3.578077881|e=-19}} J × 683 lm/W = {{val|2.443827192723|e=-16}} lm s

This implies:
* 1 [[Lumen (unit)|lumen]], lm, = {{sfrac|{{val|e=6}}|{{val|2.246520349221536260971}}}} <math>Q_\mathrm v</math> {{math|Δ''ν''<sub>Cs</sub>}}
* 1 [[candela]], cd, = 1 lm/sr = {{sfrac|{{val|e=6}}|{{val|2.246520349221536260971}}}} <math>Q_\mathrm v</math> {{math|Δ''ν''<sub>Cs</sub>}}/sr
* 1 [[lux]], lx, = 1 lm/m<sup>2</sup> = {{sfrac|{{val|8.9875517873681764|e=2}}|{{val|1.898410313566852566340456048807087002459}}}} <math>Q_\mathrm v</math> {{math|Δ''ν''<sub>Cs</sub>}}/{{math|Δ''λ''<sub>Cs</sub>}}<sup>2</sup>

=== Summary ===

The parameters of the caesium-133 hyperfine transition radiation expressed exactly in SI units are:
* Frequency = 9,192,631,770 Hz
* Time period = {{sfrac|1|9,192,631,770}} s
* Wavelength = {{sfrac|299,792,458|9,192,631,770}} m
* Photon energy = {{val|6.09110229711386655|e=-24}} J
* Photon mass equivalent = {{sfrac|{{val|6.09110229711386655|e=-40}}|{{val|8.9875517873681764}}}} kg

If the seven base units of the SI are expressed explicitly in terms of the SI defining constants, they are:
* 1 second = {{sfrac|9,192,631,770|{{math|Δ''ν''<sub>Cs</sub>}}}}
* 1 metre = {{sfrac|9,192,631,770|299,792,458}} ''c''/{{math|Δ''ν''<sub>Cs</sub>}}
* 1 kilogram = {{sfrac|{{val|8.9875517873681764|e=40}}|{{val|6.09110229711386655}}}} ''h'' {{math|Δ''ν''<sub>Cs</sub>}}/''c''<sup>2</sup>
* 1 ampere = {{sfrac|{{val|e=9}}|{{val|1.472821982686006218}}}} ''e'' {{math|Δ''ν''<sub>Cs</sub>}}
* 1 kelvin = {{sfrac|{{val|13.80649}}|{{val|6.09110229711386655}}}} ''h'' {{math|Δ''ν''<sub>Cs</sub>}}/''k''
* 1 mole = {{val|6.02214076|e=23}} elementary entities
* 1 candela = {{sfrac|{{val|e=11}}|{{val|3.82433969151951648163130104605}}}} ''h'' {{math|Δ''ν''<sub>Cs</sub>}}<sup>2</sup> ''K''<sub>CD</sub>/sr

Ultimately, 6 of the 7 base units (all but the dimensionless mole) notably have values that depend on that of {{math|Δ''ν''<sub>Cs</sub>}}, which appears far more often than any of the other defining constants.  However, the derived unit of one coulomb, which is an ampere-second, is a dimensionful unit defined purely in terms of the elementary charge and hence is independent of {{math|Δ''ν''<sub>Cs</sub>}}.